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User blog:Rgetar/Idea of program
I'm going to write a program to automatically create lists of ordinals such as this. Few weeks ago I already tried to write such a program, but it computed fundamental sequence elements only up to ε0 (by the way, I used that program to make computations for a comment to this blog). Now I'm going to make my new program compute fundamental sequence elements of ordinals up to BHO. I'll use such format for ordinals: Cantor normal form without coefficients, 0 for 0, (X) for generalized Veblen function φ(X), for array X pairs element, separated with ",", and if coordinates = 0, then is omitted. So, ordinals up to BHO are represented using 8 symbols: 0, 1, +, ,, (, ), <, >. Examples: 0 = 0 1 = 1 2 = 1+1 5 = 1+1+1+1+1 ω = (1) ω + 1 = (1)+1 ω2 = (1)+(1) ω2 = (1+1) ω6 = (1+1+1+1+1+1) ωω = ((1)) ωωω = (((1))) ε0 = (<1>1) ε1 = (<1>1,1) εω = (<1>1,(1)) ζ0 = (<1>1+1) φ(ω, 0) = (<1>(1)) Γ0 = (<1+1>1) φ(1, 0, 0, 0) = (<1+1+1>1) SVO = (<(1)>1) LVO = (<<1>1>1) etc. (But after creating a list I'm going to transform ordinals to more "common" form). For now I made transformation of an ordinal to its standard form (this was the most difficult part, but this is necessary, since for non-standard forms (in any of this and this fundamental sequence systems) fundamental sequences differ from fundamental sequence for standard form and from each other). Example of different fundamental sequences for standard and non-standard form of ordinal: Standard form of ε0 is ε0 = φ(1, 0), and its non-standard form is ωε0 = φ(φ(1, 0)). Fundamental sequence of standard form: ε00 = 0 ε01 = 1 ε02 = ω ε03 = ωω ε04 = ωωω ε05 = ωωωω Fundamental sequence of non-standard form: ωε00 = 1 ωε01 = ω ωε02 = ωω ωε03 = ωωω ωε04 = ωωωω ωε05 = ωωωωω I'll use my 6th fundamental sequence system. Left to do: computation of fundamental sequence elements and generating lists. And, I hope, there will not be big problems. Update: computation of fundamental sequence elements of 6th fundamental sequence system done. Now I'm going to do "modified" 6th fundamental sequence system. But for now I don't know how to do it. I'll try to do this: if X - successor ordinal, then φ(X)n = φ(X−1)⋅(n + 1) instead of φ(X)n = φ(X−1)⋅n; and if leo(X) = 0, lbeo(X0) = 1, lrt(X0) = 0, then δ = 1 (instead of δ = 0). Update: generating lists done, but still in forementioned format. For ε0 at level 4 there are 54 fundamental sequence: 53 fundamental sequences from this list and one extra fundamental sequence between FS #30 and FS #31 due to slightly different fundamental sequence system: ωω2 + ωω ωω2 + ωω2 ωω2 + ωω3 ωω2 + ωω4 ωω2 + ωω5 ωω2 + ωω6 ωω2 + ωω7 ωω2 + ωω8 ωω2 + ωω9 ωω2 + ωω10 Number of fundamental sequences for ε0: level 1 - 2 level 2 - 5 level 3 - 15 level 4 - 54 level 5 - 235 level 6 - 1244 level 7 - 8049 level 8 - 63877 level 9 - > 144836 (could not find due to out of memory, estimate: no more than ~ 1200000, probably few hundreds of thousands) Left to do: test up to BHO, then convert ordinals to form with non-1 coefficients, ω, ε, ζ, η, φ, Γ. Update: generating lists doesn't work beyond ε0 due to error in comparison of ordinals. Update: works. Now I'm trying to make the program work faster. Update: acceleration done. Need more acceleration. Update: apparently, comparison of ordinals (and therefore finding standard form of ordinal) cannot be significantly accelerated. But it turned out that there is no need to find standard form of every fundamental sequence element. It is enough to find standard form of 0-th element of fundamental sequence (at least for the system which I use), other elements are already in standard form, if standard form of φ(α) (for φ(α + 1), where α is ordinal) and of δ is used. Number of fundamental sequences for BHO: level 1 - 2 level 2 - 6 level 3 - 22 level 4 - 134 level 5 - 4594 level 6 - > 250000 (estimate: no more than ~ 160000000) Now not modified 6th fundamental sequence system. Numbers of fundamental sequences for ε0 are similar to numbers, which I got by hand, but shifted by one level due to the fact that in the program I add 0 to the list in the beginning: level 1 - 1 level 2 - 2 level 3 - 5 level 4 - 14 level 5 - 44 level 6 - 155 level 7 - 614 level 8 - 2742 level 9 - 13834 level 10 - 79050 Number of fundamental sequences for BHO: level 1 - 2 level 2 - 6 level 3 - 26 level 4 - 178 level 5 - 1931 level 6 - 31747 Update: I made "original" modified 6th fundamental sequence system. For ε0 I got the same lists as lists I got by hand. Number of fundamental sequences for ε0: level 1 - 2 level 2 - 5 level 3 - 15 level 4 - 53 level 5 - 221 level 6 - 1095 level 7 - 6475 level 8 - 45792 For BHO I got lists different from lists I got by hand. The reason is that in the program I use modified 6th fundamental sequence system, and in that list I used 2nd fundamental sequence system. At least I thought so. It turned out, that in that list I used not 2nd, but a bit different fundamental sequence system. I named it modified "clested" 2nd fundamental sequence system. I expected that due to the different fundamental sequence systems at level 4 should be 193 fundamental sequences (actually, this is not for modified 6th system, but for another system. which I named modified "clested" 6th system) instead of 207. Specifically, instead of 8 fundamental sequences from #149 to #156 should be 1 fundamental sequence ωΓ0 + 1 φ(ωΓ0 + 1, 0) φ(φ(ωΓ0 + 1, 0), 0) φ(φ(φ(ωΓ0 + 1, 0), 0), 0) φ(φ(φ(φ(ωΓ0 + 1, 0), 0), 0), 0) φ(φ(φ(φ(φ(ωΓ0 + 1, 0), 0), 0), 0), 0) φ(φ(φ(φ(φ(φ(ωΓ0 + 1, 0), 0), 0), 0), 0), 0) φ(φ(φ(φ(φ(φ(φ(ωΓ0 + 1, 0), 0), 0), 0), 0), 0), 0) φ(φ(φ(φ(φ(φ(φ(φ(ωΓ0 + 1, 0), 0), 0), 0), 0), 0), 0), 0) φ(φ(φ(φ(φ(φ(φ(φ(φ(ωΓ0 + 1, 0), 0), 0), 0), 0), 0), 0), 0), 0) ... Γ1 and instead of 8 fundamental sequences from #187 to #194 should be 1 fundamental sequence ωφ(⟨1, 0⟩1) + 1 φ(⟨ωφ(⟨1, 0⟩1) + 1⟩1) φ(⟨φ(⟨ωφ(⟨1, 0⟩1) + 1⟩1)⟩1) φ(⟨φ(⟨φ(⟨ωφ(⟨1, 0⟩1) + 1⟩1)⟩1)⟩1) φ(⟨φ(⟨φ(⟨φ(⟨ωφ(⟨1, 0⟩1) + 1⟩1)⟩1)⟩1)⟩1) φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨ωφ(⟨1, 0⟩1) + 1⟩1)⟩1)⟩1)⟩1)⟩1) φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨ωφ(⟨1, 0⟩1) + 1⟩1)⟩1)⟩1)⟩1)⟩1)⟩1) φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨ωφ(⟨1, 0⟩1) + 1⟩1)⟩1)⟩1)⟩1)⟩1)⟩1)⟩1) φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨ωφ(⟨1, 0⟩1) + 1⟩1)⟩1)⟩1)⟩1)⟩1)⟩1)⟩1)⟩1) φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨φ(⟨ωφ(⟨1, 0⟩1) + 1⟩1)⟩1)⟩1)⟩1)⟩1)⟩1)⟩1)⟩1)⟩1) ... φ(⟨1, 0⟩1, 1) Number of fundamental sequences for BHO for 8 different fundamental sequence systems: clested modified 6th level 1 - 2 level 2 - 6 level 3 - 28 level 4 - 193 level 5 - 1955 level 6 - 29283 (At levels 1 - 4 exactly as I expected). clested modified 2nd level 1 - 2 level 2 - 6 level 3 - 28 level 4 - 207 level 5 - 2475 level 6 - 49324 (At levels 1 - 4 exactly as in list I made by hand. So, in that list I made no one error, not to mention the fact that I thought this is list for modified 2nd system, but actually it is list for clested modified 2nd system). modified 6th level 1 - 2 level 2 - 6 level 3 - 29 level 4 - 241 level 5 - 3447 level 6 - 80331 modified 2nd level 1 - 2 level 2 - 6 level 3 - 28 level 4 - 216 level 5 - 2768 level 6 - 59234 clested 6th level 1 - 2 level 2 - 6 level 3 - 25 level 4 - 150 level 5 - 1292 level 6 - 15982 clested 2nd level 1 - 2 level 2 - 6 level 3 - 25 level 4 - 161 level 5 - 1693 level 6 - 31605 6th level 1 - 2 level 2 - 6 level 3 - 26 level 4 - 178 level 5 - 1931 level 6 - 31736 (I already posted this hereinabove). 2nd level 1 - 2 level 2 - 6 level 3 - 25 level 4 - 166 level 5 - 1834 level 6 - 35652 Further I'll use clested modified 6th, since it is fastest of these modified systems. And I will no longer use forementioned "pseudo-modified" 6th system, since I made it because of I did not know how to made "real" modified 6th system. Left to do: make limitation on the number of consecutive expansions, then convert ordinals. Update: limitation on the number of consecutive expansions done. It turned out that in the lists there are many small ordinals and few large ordinals. I'll try to expand more than 1 elements of fundamental sequence simultaneously, that is at the same level. Update: done. Now I'll do converting. Update: done. Converting of array X: if cofbeo(X) = 0 or (4 > cofbeo(X) > 0 and nobe(X) > 1) then X is written as list of all elements of X, separated with commas, else X is written as list of all base elements of X with theirs coordinates as left superscripts. Category:Blog posts